**AMS/MAA Textbooks**

Volume: 54;
2019;
399 pp;
Hardcover

MSC: Primary 34; 35; 44; 42;

**Print ISBN: 978-1-4704-5173-8
Product Code: TEXT/54**

List Price: $85.00

AMS Member Price: $63.75

MAA Member Price: $63.75

**Electronic ISBN: 978-1-4704-5364-0
Product Code: TEXT/54.E**

List Price: $85.00

AMS Member Price: $63.75

MAA Member Price: $63.75

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#### Supplemental Materials

# Lectures on Differential Equations

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*Philip L. Korman*

MAA Press: An Imprint of the American Mathematical Society

Lectures on Differential Equations
provides a clear and concise presentation of differential equations
for undergraduates and beginning graduate students. There is more
than enough material here for a year-long course. In fact, the text
developed from the author's notes for three courses: the undergraduate
introduction to ordinary differential equations, the undergraduate
course in Fourier analysis and partial differential equations, and a
first graduate course in differential equations. The first four
chapters cover the classical syllabus for the undergraduate ODE course
leavened by a modern awareness of computing and qualitative methods.
The next two chapters contain a well-developed exposition of linear
and nonlinear systems with a similarly fresh approach. The final two
chapters cover boundary value problems, Fourier analysis, and the
elementary theory of PDEs.

The author makes a concerted effort to use plain language and to
always start from a simple example or application. The presentation
should appeal to, and be readable by, students, especially students in
engineering and science. Without being excessively theoretical, the
book does address a number of unusual topics: Massera's theorem,
Lyapunov's inequality, the isoperimetric inequality, numerical
solutions of nonlinear boundary value problems, and more. There are
also some new approaches to standard topics including a rethought
presentation of series solutions and a nonstandard, but more
intuitive, proof of the existence and uniqueness theorem. The
collection of problems is especially rich and contains many very
challenging exercises.

Philip Korman is professor of mathematics at the University of
Cincinnati. He is the author of over one hundred research articles in
differential equations and the monograph Global Solution Curves
for Semilinear Elliptic Equations. Korman has served on the
editorial boards of Communications on Applied Nonlinear
Analysis, Electronic Journal of Differential Equations,
SIAM Review, an\ d Differential Equations and
Applications.

A solutions manual for this title is available electronically to
those instructors who have adopted the textbook for classroom use.
Please send email to textbooks@ams.org for more
information.

#### Readership

Undergraduate and graduate students interested in differential equations.

#### Reviews & Endorsements

In this introduction to differential equations, the author emphasizes that he wanted to write a book that students would like to read. While theoretical results are mentioned, stated, and occasionally proved throughout the book, the focus is on solving differential equations and understanding properties of their solutions.

-- Bill Satzer, MAA Reviews

#### Table of Contents

# Table of Contents

## Lectures on Differential Equations

- Cover Cover11
- Title page iii5
- Copyright iv6
- Contents v7
- Introduction xi13
- Chapter 1. First-Order Equations 115
- 1.1. Integration by Guess-and-Check 115
- 1.2. First-Order Linear Equations 317
- 1.3. Separable Equations 822
- 1.4. Some Special Equations 1731
- 1.5. Exact Equations 2640
- 1.6. Existence and Uniqueness of Solution 3044
- 1.7. Numerical Solution by Euler’s Method 3044
- 1.8*. The Existence and Uniqueness Theorem 4054

- Chapter 2. Second-Order Equations 4963
- 2.1. Special Second-Order Equations 4963
- 2.2. Linear Homogeneous Equations with Constant Coefficients 5468
- 2.3. The Characteristic Equation Has Two Complex Conjugate Roots 5872
- 2.4. Linear Second-Order Equations with Variable Coefficients 6478
- 2.5. Some Applications of the Theory 6781
- 2.6. Nonhomogeneous Equations 7286
- 2.7. More on Guessing of 𝑌(𝑡) 7589
- 2.8. The Method of Variation of Parameters 7791
- 2.9. The Convolution Integral 7993
- 2.10. Applications of Second-Order Equations 8195
- 2.11. Further Applications 92106
- 2.12. Oscillations of a Spring Subject to a Periodic Force 97111
- 2.13. Euler’s Equation 101115
- 2.14. Linear Equations of Order Higher Than Two 105119
- 2.15. Oscillation and Comparison Theorems 117131

- Chapter 3. Using Infinite Series to Solve Differential Equations 125139
- Chapter 4. The Laplace Transform 149163
- Chapter 5. Linear Systems ofDifferential Equations 177191
- Chapter 6. Nonlinear Systems 221235
- Chapter 7. The Fourier Series and Boundary Value Problems 259273
- 7.1. The Fourier Series for Functions of an Arbitrary Period 259273
- 7.2. The Fourier Cosine and the Fourier SineSeries 265279
- 7.3. Two-Point Boundary Value Problems 268282
- 7.4. The Heat Equation and the Method of Separation of Variables 273287
- 7.5. Laplace’s Equation 279293
- 7.6. The Wave Equation 283297
- 7.7. Calculating Earth’s Temperature and Queen Dido’s Problem 293307
- 7.8. Laplace’s Equation on Circular Domains 297311
- 7.9. Sturm-Liouville Problems 301315
- 7.10. Green’s Function 308322
- 7.11. The Fourier Transform 317331
- 7.12. Problems on Infinite Domains 319333

- Chapter 8. Elementary Theory of PDE 325339
- 8.1. Wave Equation: Vibrations of an InfiniteString 325339
- 8.2. Semi-Infinite String: Reflection of Waves 330344
- 8.3. Bounded String: Multiple Reflections 334348
- 8.4. Neumann Boundary Conditions 336350
- 8.5. Nonhomogeneous Wave Equation 339353
- 8.6. First-Order Linear Equations 346360
- 8.7. Laplace’s Equation: Poisson’s Integral Formula 352366
- 8.8. Some Properties of Harmonic Functions 354368
- 8.9. The Maximum Principle 357371
- 8.10. The Maximum Principle for the Heat Equation 359373
- 8.11. Dirichlet’s Principle 363377
- 8.12. Classification Theory for Two Variables 365379

- Chapter 9. Numerical Computations 377391
- Appendix 387401
- Bibliography 395409
- Index 397411
- Back Cover Back Cover1414